Abstract
A one-dimensional differential game is considered, in which the payoff is determined by the modulus of the deviation of the phase variable at a fixed time from the set value, taking into account the periodicity. The first player seeks to minimize the payoff. The goal of the second player is the opposite. For this problem, the price of the game is calculated and optimal player controls are constructed. As an example, we consider the problem of controlling a rotational mechanical system in which the goal of the first player acquires the meaning of minimizing the modulus of deviation of the angle from the desired state.
This work was funded by the Russian Science Foundation (project no. 19-11-00105).
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References
Andrievsky, B.R.: Global stabilization of the unstable reaction-wheel pendulum. Control Big Syst. 24, 258–280 (2009). (in Russian)
Beznos, A.V., Grishin, A.A., Lenskiy, A.V., Okhozimskiy, D.E., Formalskiy, A.M.: The control of pendulum using flywheel. In: Workshop on Theoretical and Applied Mechanics, pp. 170–195. Publishing of Moscow State University, Moscow (2009). (in Russian)
Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York (1965)
Izmest’ev, I.V., Ukhobotov, V.I.: On a linear control problem under interference with a payoff depending on the modulus of a linear function and an integral. In: 2018 IX International Conference on Optimization and Applications (OPTIMA 2018) (Supplementary Volume), DEStech, pp. 163–173. DEStech Publications, Lancaster (2019). https://doi.org/10.12783/dtcse/optim2018/27930
Izmest’ev, I.V., Ukhobotov, V.I.: On a single-type differential game with a non-convex terminal set. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 595–606. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22629-9_42
Krasovskii, N.N., Subbotin, A.I.: Positional Differential Games. Nauka Publishing, Moscow (1974). (in Russian)
Kudryavtsev, L.D.: A Course of Mathematical Analysis, vol. 1. Vysshaya Shkola Publishers, Moscow (1981). (in Russian)
Leonov, G.A.: Introduction to Control Theory. Publishing St. Petersburg University, St. Petersburg (2004). (in Russian)
Pontryagin, L.S.: Linear differential games of pursuit. Math. USSR-Sbornik 40(3), 285–303 (1981). https://doi.org/10.1070/SM1981v040n03ABEH001815
Ukhobotov, V.I.: Synthesis of control in single-type differential games with fixed time. Bull. Chelyabinsk Univ. 1, 178–184 (1996). (in Russian)
Ukhobotov, V.I., Izmest’ev, I.V.: Single-type differential games with a terminal set in the form of a ring. In: Systems dynamics and control process. In: Proceedings of the International Conference, Dedicated to the 90th Anniversary of Acad. N.N. Krasovskiy, Ekaterinburg, 15–20 September 2014, pp. 325–332. Publishing House of the UMC UPI, Ekaterinburg (2015). (in Russian)
Ushakov, V.N., Ukhobotov, V.I., Ushakov, A.V., Parshikov, G.V.: On solution of control problems for nonlinear systems on finite time interval. IFAC-PapersOnLine 49(18), 380–385 (2016). https://doi.org/10.1070/10.1016/j.ifacol.2016.10.195
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Izmest’ev, I.V., Ukhobotov, V.I. (2020). On a One-Dimensional Differential Game with a Non-convex Terminal Payoff. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_14
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